Mathematical Sciences: L-Independence in Arithmetic Algebraic Geometry

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de Jong 9625417 This proposal plans to investigate one of the fundamental questions currently being studied in arithmetic geometry, namely the invariance of certain cohomologically defined invariants using various cohomology theories, especially l-adic (etale) cohomologies for varying l and p-adic (rigid) cohomology. The questions to be asked are whether dimensions of certain spaces are independent of l, whether the eigenvalues of correspondences acting on certain groups are independent of l, whether the eigenvalues of Frobenius are independent of l, etc. It is conjectured that the answers to these questions are all positive. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.